# The inter-request time distribution after aggregating some arrivals in the renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process"

The inter-arrival time of a renewal process, $t$, conforms to a general distribution, denoted by PDF $f(t)$.

Next we aggregate the requests according to the following pattern: from the first arrival, within the fixed-length time interval $T$, the requests in this interval are aggregated to the first arrival. In other words, the requests in this interval are removed except the first one. This procedure repeats for the rest arrivals. The following figure illustrates this aggregation pattern.

My question is, what is the distribution of the inter-arrival times after the request aggregation? What if $T$ is a random variable?

If accurate distribution functions are difficult to obtain, is there any approximate ways to do this? Can I just assume that the output process has the same inter-arrival distribution with the input one, but with smaller request rate since a portion of the arrivals are removed.